chaospy 4.1.1

Numerical tool for perfroming uncertainty quantification

Chaospy is a numerical tool for performing uncertainty quantification using polynomial chaos expansions and advanced Monte Carlo methods implemented in Python.

• Documentation: https://chaospy.readthedocs.io/en/master

• Source code: https://github.com/jonathf/chaospy

• Issues: https://github.com/jonathf/chaospy/issues

• Journal article: “Chaospy: An open source tool for designing methods of uncertainty quantification”

Installation

Installation should be straight forward from PyPI:

$ pip install chaospy

Example Usage

chaospy is created to work well inside numerical Python ecosystem. You therefore typically need to import Numpy along side chaospy:

>>> import numpy
>>> import chaospy

chaospy is problem agnostic, so you can use your own code using any means you find fit. The only requirement is that the output is compatible with numpy.ndarray format:

>>> coordinates = numpy.linspace(0, 10, 100)
>>> def forward_solver(coordinates, parameters):
...     """Function to do uncertainty quantification on."""
...     param_init, param_rate = parameters
...     return param_init*numpy.e**(-param_rate*coordinates)

We here assume that parameters contains aleatory variability with known probability. We formalize this probability in chaospy as a joint probability distribution. For example:

>>> distribution = chaospy.J(
...     chaospy.Uniform(1, 2), chaospy.Normal(0, 2))
>>> print(distribution)
J(Uniform(lower=1, upper=2), Normal(mu=0, sigma=2))

Most probability distributions have an associated expansion of orthogonal polynomials. These can be automatically constructed:

>>> expansion = chaospy.generate_expansion(8, distribution)
>>> print(expansion[:5].round(8))
[1.0 q1 q0-1.5 q0*q1-1.5*q1 q0**2-3.0*q0+2.16666667]

Here the polynomial is defined positional, such that q0 and q1 refers to the uniform and normal distribution respectively.

The distribution can also be used to create (pseudo-)random samples and low-discrepancy sequences. For example to create Sobol sequence samples:

>>> samples = distribution.sample(1000, rule="sobol")
>>> print(samples[:, :4].round(8))
[[ 1.5         1.75        1.25        1.375     ]
 [ 0.         -1.3489795   1.3489795  -0.63727873]]

We can evaluating the forward solver using these samples:

>>> evaluations = numpy.array([
...     forward_solver(coordinates, sample) for sample in samples.T])
>>> print(evaluations[:3, :5].round(8))
[[1.5        1.5        1.5        1.5        1.5       ]
 [1.75       2.00546578 2.29822457 2.63372042 3.0181921 ]
 [1.25       1.09076905 0.95182169 0.83057411 0.72477163]]

Having all these components in place, we have enough components to perform point collocation. Or in other words, we can create a polynomial approximation of forward_solver:

>>> approx_solver = chaospy.fit_regression(
...     expansion, samples, evaluations)
>>> print(approx_solver[:2].round(4))
[q0 -0.0002*q0*q1**3+0.0051*q0*q1**2-0.101*q0*q1+q0]

Since the model approximations are polynomials, we can do inference on them directly. For example:

>>> expected = chaospy.E(approx_solver, distribution)
>>> print(expected[:5].round(8))
[1.5        1.53092356 1.62757217 1.80240142 2.07915608]
>>> deviation = chaospy.Std(approx_solver, distribution)
>>> print(deviation[:5].round(8))
[0.28867513 0.43364958 0.76501802 1.27106355 2.07110879]

For more extensive guides on this approach an others, see the tutorial collection.

Contributions and Related Projects

A few shout-outs to people who have contributed to the Chaospy project:

OpenTURNS	Thanks to Régis Lebrun for both proposing a collaboration and creating an initial implementation of both Chaospy Compatibility in OpenTURNS and OpenTURNS Compatibility in chaospy.
orthopy quadpy	Thanks to Nico Schlömer for providing the implementation for several of the quadrature integration methods.
UQRF	Thanks to Florian Künzner for providing the initial implementation of kernel density estimation and quantity-of-interest distribution.

Chaospy is being used in other related projects that requires uncertainty quantification components chaospy provides. For example:

easyVVUQ	Library designed to facilitate verification, validation and uncertainty quantification.
STARFiSh	Shell-based, scientific simulation program for blood flow in mammals.
Profit	Probabilistic response model fitting via interactive tools.
UncertainPy	Uncertainty quantification and sensitivity analysis, tailored towards computational neuroscience.
SparseSpACE	Spatially adaptive combination technique targeted to solve high dimensional numerical integration.

For a more comprehensive list, see Github’s dependency graph.

Questions and Contributions

Please feel free to file an issue for:

• bug reporting

• asking questions related to usage

• requesting new features

• wanting to contribute with code

If you are using this software in work that will be published, please cite the journal article: Chaospy: An open source tool for designing methods of uncertainty quantification.

And if you use code to deal with stochastic dependencies, please also cite Multivariate Polynomial Chaos Expansions with Dependent Variables.